Damped Gyroscopic Systems Research Articles

Uncoupling of Damped Linear Potential Multi-degree-of-freedom Structural and Mechanical Systems

Abstract This paper provides the necessary and sufficient conditions for a multi-degree-of-freedom linear potential system with an arbitrary damping matrix to be uncoupled into independent subsystems of at most two degrees-of-freedom using a real orthogonal transformation. The incorporation of additional information about the matrices, which many structural and mechanical systems commonly possess, shows a reduction in the number of these conditions to three. Several new results are obtained and are illustrated through examples. A useful general form for the damping matrix is developed that guarantees uncoupling of such systems when they satisfy just two conditions. The results provided herein lead to new physical insights into the dynamical behavior of potential systems with general damping matrices and to robust computational procedures. It is shown that the dynamics of a damped potential system in which the damping matrix may be arbitrary is identical to that of a damped gyroscopic potential system with a symmetric damping matrix. This brings, for the first time, these two systems, which are seen today as belonging to different categories of dynamical systems, under a unified framework.

Semi-analytical solutions for forced and free vibration of damped fluid-conveying pipe systems based on complex modal superposition method

In this paper, the response solutions of the damped fluid-conveying pipe system with elastic torsion constraints at both ends are analyzed. The pipe system considering gyroscopic effect induced by internal flow and damping effect is a typical damped gyroscopic system. This system cannot be decoupled in the modal space by the traditional modal analysis, and then the semi-analytical response solutions cannot be obtained by the classical modal superposition method. In order to remedy this problem, this paper proposes a new method based on complex modal superposition method and state-space method to give the semi-analytical response solutions of the damped fluid-conveying pipe system. The adjoint system is introduced through the concept of adjoint operator, and the orthogonality conditions of the damped fluid-conveying pipe system are derived in the state-space by using the eigenvalue relationship between the original system and the adjoint system. The Laplace transformation method is used to solve the eigenvalue problems of the original system and the adjoint system, and the implicit function equations about the eigenvalues and analytical eigenfunctions are obtained. According to the obtained orthogonality conditions, the semi-analytical response solutions of the system under arbitrary initial conditions and excitations are given, and the obtained solutions include transient and steady-state response solutions. In the numerical discussion part, the reliability and accuracy of the present method are verified. This research has positive significance for the dynamic analysis of fluid-conveying pipe systems, and the obtained orthogonality conditions have potential value for the nonlinear dynamics research.

Quadratic model updating for damped gyroscopic systems

This paper is concerned with the problem of the optimal approximation for a given matrix pencil (Ma, Da, Ga, Ka, Na ) under the spectral constraint and the symmetric constraint. Such a problem arises in finite element model updating for damped gyroscopic systems. By using constrained optimization theory and matrix derivatives, an explicit formulation for the solution of the problem is established. The efficiency and accuracy of the proposed method is numerically verified by a simple five-degree-of-freedom system.

An inverse quadratic eigenvalue problem for damped gyroscopic systems

ABSTRACT Model updating for damped gyroscopic systems can be mathematically formulated as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximation problem (OAP). Problem IQEP: Given , where p<n, and both Λ and X are closed under complex conjugation. Find real symmetric matrices and a real skew-symmetric matrix G (that is, ) such that Problem OAP: Given symmetric matrices and a skew-symmetric matrix , find such that where is the solution set of Problem IQEP. The representation of the general solution of Problem IQEP is deduced and the optimal approximation solution of Problem OAP is obtained.

Reduced-order modeling for rotating rotor-bearing systems with cracked impellers using three-dimensional finite element models

A novel reduced-order modeling method is presented in this paper for dynamics analysis of rotating impeller–shaft-bearing assembly with cracked impellers. Based on three-dimensional finite element model, the complex component mode synthesis (CMS) method is employed to generate an efficient reduced-order model (ROM) for studying the effects of crack on the global vibration of the rotating assembly. First, a modeling framework for impeller–shaft-bearing systems in rotating frame is presented. Rotational effects, including Coriolis matrix and centrifugal softening, have been taken into account. Then, the governing equation of motion of the damped gyroscopic system is reduced by the complex CMS method. Finally, the obtained ROM is employed to study the effects of crack on assembly׳s vibration. During the steady-state response analysis, external excitations on the impeller due to rotor–stator interactions have been taken into account, which was however neglected in previous investigations on rotordynamics. Numerical results show that the lower-order eigenvalues and the unbalance response of the assembly are not sensitive to the local crack on impeller. Nevertheless, the flexible coupling between impeller and shaft becomes more complex when the air flow-induced excitations are considered. Under EO1 traveling wave excitations, a crack leads to slight changes in the assembly׳s response. In contrast, the effect of crack becomes significant when the assembly is excited by EO2 and higher EO excitations. Moreover, the nonlinear crack breathing effects affect the assembly׳s response obviously. Finally, a potential technique for detecting the crack on impeller during operation is discussed.

Quadratic inverse eigenvalue problem for damped gyroscopic systems

In this paper, we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems, that is, we are to construct a damped gyroscopic second order system having given 2 n eigenvalues a...

Finite Element Method Applied to the Eigenvalue Analysis of Flexible Rotors Supported by Journal Bearings

This work deals with a finite element procedure developed to perform the eigenvalue analysis of damped gyroscopic systems, represented by flexible rotors supported on fluid film journal bearings. The rotor finite element model is based on the Timoshenko beam theory, accounting for the shaft rotary inertia and gyroscopic moments. The governing equations for the hydrodynamic journal bearing are obtained through the Galerkin weighted residual method applied to the classical Reynolds equation. A perturbation scheme on the fluid film governing equation permits to obtain the zero-th and first order lubrication equations for the bearings, which allow the computation of the dynamic force coefficients associated with the bearing stiffness and damping. The rotor-bearing system equation, which consists of a case of damped gyroscopic equation, is rewritten on state form to compute the complex eigenvalues. The natural frequencies at several operating conditions are obtained and compared to the technical literature data. The influence of the effective damping on the eigenvalue real part sign is analyzed for some examples of rotor-bearing systems, showing how the stability conditions can be predicted by the eigenvalue analysis. The procedure implemented in this work can provide useful guidelines and technical data about the selection of the more appropriate set of bearing parameters for rotating machines operating at stringent conditions.

Quadratic inverse eigenvalue problem for damped gyroscopic systems

In this paper, we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems, that is, we are to construct a damped gyroscopic second order system having given 2n eigenvalues and associated eigenvectors as its full eigenstructure. Under some mild assumptions, the coefficient matrices are constructed in a recursive process. We also discuss how to obtain a damped gyroscopic system with the leading coefficient matrix M being positive definite.

A direct updating method for damped gyroscopic systems using measured modal data

The matrix model updating problem (MMUP), considered in this paper, concerns updating a symmetric second-order finite element model so that the updated model can reproduce a given set of measured eigenvalues and eigenvectors by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. The MMUP can be mathematically formulated as following two problems. Problem 1: Given M a ∈ R n × n , Λ = diag { λ 1 , … , λ p } ∈ C p × p , X = [ x 1 , … , x p ] ∈ C n × p , where p < n and both Λ and X are closed under complex conjugation in the sense that λ 2 j = λ ¯ 2 j - 1 ∈ C , x 2 j = x ¯ 2 j - 1 ∈ C n for j = 1 , … , l , and λ k ∈ R , x k ∈ R n for k = 2 l + 1 , … , p , find real-valued symmetric matrices D, K and a real-valued skew-symmetric matrix G (that is, G T = - G ) such that M a X Λ 2 + ( D + G ) X Λ + KX = 0 . Problem 2: Given real-valued symmetric matrices D a , K a ∈ R n × n and a real-valued skew-symmetric matrix G a , find ( D ^ , G ^ , K ^ ) ∈ S E such that ‖ D ^ - D a ‖ 2 + ‖ G ^ - G a ‖ 2 + ‖ K ^ - K a ‖ 2 = min ( D , G , K ) ∈ S E ( ‖ D - D a ‖ 2 + ‖ G - G a ‖ 2 + ‖ K - K a ‖ 2 ) , where S E is the solution set of Problem 1 and ‖ · ‖ is the Frobenius norm. We provide the representation of the general solution of Problem 1 and show that the optimal approximation solution ( D ^ , G ^ , K ^ ) is unique and derive an explicit formula for it.

The direct updating of damping and gyroscopic matrices

In this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems. This model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data.

Universal Matrix Perturbation Method for Structural Dynamic Reanalysis of General Damped Gyroscopic Systems

We investigate an effective matrix perturbation method for structural dynamic reanalysis of general damped gyroscopic systems. By using the complex eigensubspace condensation and the or thogonal decomposition procedures, two greatly reduced generalized eigenvalue equations are obtained. The lower-order perturbations of eigensolutions (i.e. complex eigenvalues and the corresponding left and right eigenvectors) are then determined by solving the two reduced eigenvalue problems. The higher-order perturbations of eigensolutions are obtained by executing a singular value decomposition procedure for a complex matrix. The proposed method is a universal perturbation method, for it is universally applicable to the reanalysis of general damped gyroscopic systems with all three cases of complex eigenvalues: distinct, repeated, and closely spaced eigenvalues. Numerical examples corresponding to the three different cases of eigenvalues are presented. The perturbed eigensolutions are computed using the present method and compared with the exact solutions.

Perturbation Technique for Non-Self-Adjoint Systems with Repeated Eigenvalues

The reanalysis of non-self-adjoint systems is computationally very expensive in the design of structural dynamics. An effective, even if approximate, evaluation of the eigensolutions of a modified system is valuable to designers. In a paper by Meirovitch and Ryland (Meirovitch, L., and Ryland, G., A Perturbation Technique for Gyroscopic Systems with Small Internal and External Damping,' Journal of Sound and Vibration, Vol. 100, No. 3, 1985, pp. 393408), a perturbation technique was used to derive the perturhed solutions of a damped gyroscopic system with distinct eigenvalues. That approach is extended to the case of systems with repeated eigenvalues. A numerical example is presented to verify the present method, and satisfactory results are observed.

A reformulated Arnoldi algorithm for non-classically damped eigenvalue problems

In applying Arnoldi method to non-symmetric eigenvalue problems for damped structures, a structure of the projected upper Hessenberg matrix is obtained in this paper. By exploiting the structure of the upper Hessenberg matrix and taking advantages of the block properties of system matrices, the Arnoldi reduction algorithm is reformulated for less computation and higher accuracy. In conjunction with the reformulated Arnoldi algorithm, real Schur decomposition instead of Jordan decomposition is adopted aiming at non-complex arithmetic, non-discriminative processing of defective and non-defective systems and numeric stability. A concise reduction algorithm for eigenproblems for undamped gyroscopic systems is obtained by directly degenerating from the reformulated Arnoldi algorithm. For safely solving engineering problems without omitting eigenvalues, a restart reduction procedure is proposed in terms of the reformulated reduction algorithm with deflation developed in this paper. Numerical examples once solved with algorithms originated from Lanczos methods were re-solved. In addition, the non-symmetric eigenvalue problem for a shear wall by BEM modeling and a damped gyroscopic system with eigenvalues of high multiplicity were also used to demonstrate the efficacy of the presented methods. © 1997 John Wiley & Sons, Ltd.

Eigenvalue and Eigenvector Determination for Damped Gyroscopic Systems

We derive and examine a new method that uses the undamped gyroscopic system as the unperturbed system. The accuracy and convergence of the new method are independent of the size of the gyroscopic terms. Futhermore, the method permits a numerically efficient solution of the eigen problem for damped gyroscopic systems because it can use the approach of Meirovitch (1974) to solve the eigenproblem for the unperturbed system

The eigenvalue problem for damped gyroscopic systems

This paper develops the dynamic orthogonal decoupling method in order to decouple damped gyroscopic structural dynamic systems. The Arnoldi method is extended to solve the generalised eigenvalue problem via a real Schur decomposition. Very interesting results are obtained, which show that this method cannot only obtain the eigenvalues of the system, but can also solve dynamic response of damped gyroscopic dynamic systems in a decoupled way. Compared to the conventional complex mode decoupling method, this method virtually uses only real number computation, remains valid when the corresponding eigenvalue problem is defective and does not require solution of the left eigenvector problem. The structure of the decoupled dynamic equation is very satisfying, because it enhances understanding of the combination of damping with gyroscopic effects. For undamped gyroscopic systems, the decoupling method proposed degenerates to that of Meirovitch.

The analytical approach for large scale non-classically damped dynamic system

This paper gives a dynamic decoupling approach for the analysis of large scale non-classically damped system, in which the complex variable computations were completely avoided not only in solving for the eigenvalue problem but also in the calculation of the dynamic response. The analytical approaches for undamped gyroscopic system, non-classically damped system, including the damped gyroscopic system were unified. Very interesting and useful theoretical results, practical algorithms were obtained which are applicable to both non-defective and defective systems.

Comment on 'Modal analysis of structures with holonomic constraints'

I the paper under consideration Yang studied the influence of the introduction of additional constraints in damped gyroscopic mechanical systems. In particular, the author discussed the influence of the constraints on the system's stability behavior. Several theorems were formulated and interesting results were obtained. Unfortunately the term stable system is not clearly defined in the paper, and it is used with different meanings in different contexts. One of the author's conclusions is that for a stable, initially unconstrained, free vibration system, if the added constraints is scleronomic, the constraint system remains stable. Here is a simple counterexample that we borrow from Ref. 2. Consider the point mass elastically supported in a rigid ring rotating with circular frequency £2 according to Fig. la. Its equations of motion are

A perturbation technique for gyroscopic systems with small internal and external damping

The response of linear damped gyroscopic systems can be obtained by means of techniques of linear systems theory, which involves the computation of the transition matrix. The response is in terms of complex quantities, which is likely to cause computational difficulties as the order of the system increases. In the absence of damping, it is possible to derive the response of a linear gyroscopic system with relative ease by working with real quantities alone. When damping is small, one can use a perturbation approach to produce the response by regarding the undamped gyroscopic system as the unperturbed system. In a previous paper, a perturbation analysis was used to derive the response of a gyroscopic system with small internal damping. This paper extends the approach to the case of external damping, which is characterized not only by symmetric coefficients multiplying velocities but also by skew symmetric coefficients multiplying displacements, where the latter terms are known as circulatory. A numerical example is presented.

Optimal Control of Damped Flexible Gyroscopic Systems

An optimal control scheme for damped gyroscopic systems is presented. The procedure works with real quantities alone. The optimal control scheme is based on the concept of independent modal-space control, leading to a set of independent second-order matrix Riccati equations. The solution of such equations can be obtained with relative ease for both the steady-state and the transient case. Damping is shown to be beneficial, resulting in enhanced control with less control effort.

Response of slightly damped gyroscopic systems

A second-order perturbation theory is developed for the response of slightly damped gyroscopic systems. The solution is based on the eigensolution for undamped gyroscopic systems and is expressed in terms of real quantities alone.